This paper establishes, first, the local theory of neutral functional differential equations with infinite delay on the space of continuous and bounded functions under some Carathéodory conditions, which is complementary to Z. C. Wang and J. H. Wu [Funkc. Ekvacioj, Ser. Int. 28, 157-170 (1985; Zbl 0553.34044)], J. H. Wu [Chin. Ann. Math., Ser. A 7, 189-200 (1986; Zbl 0589.34050)]. Next, for general Volterra neutral integrodifferential equations, considering the characteristic property of Lyapunov direct method, this paper introduces a concept of stable D-operator with respect to a wedge to obtain some stability sufficient conditions. As an application to retarded FDEs, this paper weakens some stability conditions of T. A. Burton and W. E. Mahfoud [Trans. Am. Math. Soc. 279, 143-174 (1983; Zbl 0522.45006)].